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User blog:Rgetar/Square brackets and I ordinal notation
This notation is like notation from my blog Square brackets OCF. Purpose of this notation: I'll try to use this notation to make conversion into standard form a bit easier. Fundamental sequence and cofinality Definition of fundamental sequence I use: fundamental sequence of ordinal α is strictly increasing sequence of ordinals of minimal length such as α is least ordinal larger than all elements of this sequence. (Or, we can say that α is least strict upper bound of its fundamental sequence. Least strict upper bound is defined similarly to least upper bound (supremum), but "≥" is replaced with ">"). Definition of cofinality I use: cofinality of ordinal α is length of its fundamental sequence. Elements of fundamental sequence of ordinal α are enumerated using ordinals, beginning from 0: α0, α1, α2, α3, ... Designation of cofinality of ordinal α: cof(α). So, cofinality of 0 is 0, and fundamental sequence of 0 is empty set; cofinality of successor ordinal α is 1, and fundamental sequence of successor ordinal α contains only one element α0 such as α0 + 1 = α. incard(α) incard means "infinite cardinality". Definition of incard(α): :if α < ω then incard(α) = ω; :if α ≥ ω then incard(α) = card(α), where card(α) is cardinality of α. Notation Symbols This notation uses 3 symbols: :[ :] :I Strings This notation uses 3 sorts of strings: :empty string :I :βγ, where β and γ are any strings in this notation. Correspondence "string - ordinal" :Empty string is 0. :I is least weakly inaccessible cardinal. :[]γ is successor of γ: []γ = γ + 1. :If γ < I then Iγ is least uncountable cardinal larger than γ. For other strings this correspondence is defined using fundamental sequences (see next section). Cofinality and elements of fundamental sequence α is empty string (that is α = 0). cof(α) = 0. Fundamental sequence of α does not contain elements. α = βγ, cof(β) = 0 (that is β = 0, α = []γ). cof(α) = 1. α0 = γ. α = βγ, cof(β) = 1. cof(α) = ω. α0 = [β0]γ, α+ 1 = [β0]αn. α = βγ, ω ≤ cof(β) ≤ incard(γ). cof(α) = cof(β). αn = [βn]α. α = βγ, incard(γ) < cof(β) < I. cof(α) = ω. αn = [δ+ 1]γ, where δ+ 1 = [β[δn]]δn, δ0 is largest cardinal less than cof(β). α = βγ, γ < I, cof(β) = I. cof(α) = Iγ. αn = [βn]α. α = I. cof(α) = I. αn = n. Calculation of incard(α) :α is empty string (that is α = 0). incard(α) = [[]] (that is incard(α) = ω). :α = βγ, β < I, γ < I. incard(α) = incard(γ). :α = βγ, β ≥ I, γ < I. incard(α) = α. :α = βγ, γ ≥ I. incard(α) = I. Calculation of δ0 Calculation of δ0 from "5." of section "Cofinality and elements of fundamental sequence": cof(β) > [[]] = ω, since ω is least possible incard, but incard(γ) < cof(β); cof(β) ≠ I, so, cof(β) should be calculated using "4." or "6." of section "Cofinality and elements of fundamental sequence", but in "4." cofinality is calculated using cofinality (cof(α) = cof(β)), so, cof(β) should be calculated using "6." of section "Cofinality and elements of fundamental sequence": cof(α) = Iγ (here β is α); δ0 is incard of this γ: δ0 = incard(γ). Comparison of strings :empty string < I :γ < βγ :if β1 < β2 then β1γ < β2γ :if γ < I then βγ < I Note: these rules are not enough for comparison of any β1γ1 and β2γ2. Category:Blog posts